Timeline for Global section of universal bundle on Grassmanian
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 12, 2016 at 13:27 | vote | accept | Raffaele C | ||
| Jun 12, 2016 at 9:13 | comment | added | Andreas Cap | continuation: $P$ consists of block-upper-triangular matrices, so there is an obvious homomorphism $P\to S(GL(k,\mathbb C)\times GL(n-k,\mathbb C))$, Via this you can use the standard representation of $GL(k,\mathbb C)$ and its dual to induce the tautological bundle and its dual. | |
| Jun 12, 2016 at 9:11 | comment | added | Andreas Cap | You should have $U_{k,m}\times \mathbb C^k$ in your description. To get the dual bundle, you simply replace $\mathbb C^k$ by its dual space and use the canonical action, i.e.~$(g^{-1}\cdot\lambda)(v)=\lambda(g\cdot v)$. However, the description you use is not very well suited to the description of general homogeneous bundles. The better point of view is to have $G=SL(n,\mathbb C)$ and $P\subset G$ the stabilizer of $\mathbb C^k\subset\mathbb C^n$. Then for any representation $W$ of $P$, $G\times_PW$ is the quotient of $G\times W$ by $(g,w)\sim (gh,h^{-1}\cdot w)$ for $h\in P$. | |
| Jun 10, 2016 at 20:59 | comment | added | Raffaele C | I'm trying to figure out how tautological bundle and its dual can be defined on the Grassmannian in this context. If you see the Grassmannian as the quotient of the space $U_{k,m}$ of matrixes $m \times k$ ($m \geq k$) of rank $k$ modulo the right multiplication of $GL_k$, I would define the tautological bundle as the quotient of $U_{k,m} \times \mathbb{C}$ modulo the relation $(M,v) \sim (Mg, g^{-1}v)$ for any $g \in GL_k$. But then I can't see what its dual is. | |
| Jun 9, 2016 at 12:54 | comment | added | Andreas Cap | The basic story is that the compact homogeneous spaces of complex semisimple groups are exactly the quotients by parabolic subgroups, which are known as generalized flag manifolds. Any parabolic subgroup has a natural reductive quotient via homogeneous vector bundles representations of this so-called Levi-factor give rise to representations of the initial group. In my opinion, the first chapters of the book by Baston and Eastwood I mentioned give a nice introduction to the topic. The paper you mention contains an exposition of Bot-Borel-Weil, but not much background. | |
| Jun 9, 2016 at 11:12 | comment | added | Raffaele C | I found a nice paper "Grassmanians and representations" here: arxiv.org/pdf/math/0507482v2.pdf | |
| Jun 9, 2016 at 8:41 | comment | added | Raffaele C | I never saw Grassmanians under the point of view of representations but it seems powerful. Would you be able to give a brief introduction or a basic reference to understand this point of view? | |
| Jun 4, 2016 at 9:30 | history | answered | Andreas Cap | CC BY-SA 3.0 |